The rainbow

After the rain

When the Sun comes out after rain, you can sometimes see a bow of colours in the sky. This is a rainbow. It is a natural spectrum, and it is made by millions of small drops of water in the air.

What does a raindrop do?

A raindrop is a tiny sphere of water. Sunlight does three things to it, one after the other.

1. Refraction and dispersion at the first surface. The ray goes into the drop, and it is split into colours, because water bends violet more than red.
2. Internal reflection at the back of the drop. The light is reflected from the inner surface.
3. Refraction again as the light leaves the drop. This bends each colour a little more, and the colours spread.

Each drop acts like a small prism. It sends a different colour to the observer's eye from a different angle.

Why a bow?

Each colour leaves the drops at a certain angle to the direction of the sunlight. For red, this angle is about $42^\circ$. For violet, it is about $40^\circ$. An observer sees red light from the drops that are at $42^\circ$ from the line that points away from the Sun. These drops are on a circle in the sky, so the rainbow is an arc of a circle. The violet comes from drops on a smaller circle, at $40^\circ$.

So the rainbow has red on the outside and violet on the inside.

You should find that each curve has a flat top. Rays that hit the drop over a wide band all come back near that angle, so the light piles up there. Red piles up at about $42^\circ$ and violet at about $40.6^\circ$: two neighbouring rings of colour.

When do we see a rainbow?

• The Sun must be behind the observer, and the rain (or the spray) must be in front. A rainbow is never seen in the same side of the sky as the Sun.
• The Sun must be low in the sky. The higher the Sun, the lower the bow; when the Sun is high, the bow sinks out of sight.
• A waterfall or a garden hose with a fine spray can make a rainbow too.

Sometimes there is a second, fainter rainbow outside the first. In this one, the light is reflected twice inside each drop, and the colours are in the reverse order: red is on the inside.

A worked example

Late in the afternoon the Sun is $18^\circ$ above the horizon. How high is the top of the red bow?

1. The point opposite the Sun. It is on the line from the Sun through your head, so it is $18^\circ$ below the horizon.
2. The bow. The red bow is a circle $42^\circ$ around that point.
3. The top. $42^\circ - 18^\circ = 24^\circ$ above the horizon.

Practise with new numbers

Try it in Python

The program finds the angle of the rainbow for red and violet light, using the refractive index of water for each colour.

Remember these

• A rainbow is a spectrum made by raindrops: refraction with dispersion, one internal reflection, and refraction again.
• Red is on the outside, at about $42^\circ$. Violet is on the inside, at about $40^\circ$.
• The Sun is behind the observer, and the rain is in front.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Why is a rainbow curved?

2. In the fainter second bow, which colour is on the outside?

3. Why does a raindrop split sunlight into colours at all?

4. The Sun is $20^\circ$ above the horizon. How high is the top of the red bow?